1.11.40 problem 41 and 42

Internal problem ID [361]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 41 and 42
Date solved : Tuesday, September 30, 2025 at 03:57:47 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }-2 y&=8 x^{5} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=0 \\ y^{\prime \prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 47

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)-diff(y(x),x)-2*y(x) = 8*x^5; 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -4 x^{5}+10 x^{4}+20 x^{3}+30 x^{2}-450 x +255-96 \cos \left (x \right )+288 \sin \left (x \right )-160 \,{\mathrm e}^{-x}+{\mathrm e}^{2 x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

ode=D[y[x],{x,4}]-D[y[x],{x,3}]-D[y[x],{x,2}]-D[y[x],{x,1}]-2*y[x]==8*x^5; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -4 x^5+10 x^4+20 x^3+30 x^2-450 x-160 e^{-x}+e^{2 x}+288 \sin (x)-96 \cos (x)+255 \end{align*}

✓ Sympy. Time used: 0.199 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**5 - 2*y(x) - Derivative(y(x), x) - Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0, Subs(Derivative(y(x), (x, 3)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 4 x^{5} + 10 x^{4} + 20 x^{3} + 30 x^{2} - 450 x + e^{2 x} + 288 \sin {\left (x \right )} - 96 \cos {\left (x \right )} + 255 - 160 e^{- x} \]